On core of categorical product of (di)graphs
Reza Naserasr, Cyril Pujol
TL;DR
This work studies the possible order of the core of the tensor (categorical) product of graphs, a question linked to Hedetniemi-type phenomena and to the efficiency of certain database queries. It develops an orthogonality framework using $k$-bounded paths and mountains to construct large families of cores whose cone-augmented products are themselves cores, and provides a concrete sufficient condition guaranteeing this property for products of many graphs. By building a family of $\binom{2k}{k}$ digraphs with sizes $|D|$ between $k^2+5k+2$ and $3k^2+3k+2$ whose product is a core, the paper demonstrates substantial potential growth in core size. It then translates these results from digraphs to graphs using a gadget construction $G[D]$ that preserves homomorphism structure, showing the broader applicability of the approach to graph databases and related combinatorial problems.
Abstract
The core of a graph is the smallest graph (in terms of number of vertices) to which it is homomorphically equivalent. The question of the possible order of the core of the tensor product (also known as categorical, Heidetnemi or direct product) of two graphs captures some well known problems. For instance, the recent counterexample to the Hedetniemi conjecture for 5-chromatic graphs is equivalent to saying that there are cores of order at least 5 whose product has a core of order 4. In this work, motivated by a question from Leonid Libkin in the area of graph databases, we first present methods of building cores whose categorical product is also a core. Extending on this we present sufficient conditions for a set of cores to have a product which is also a core. Presenting an example of such a family of digraphs, we construct a family of $\binom{2n}{n}$ digraphs, where the number of vertices of each is between $n^2+5n+2$ and $3n^2+3n+2$ and the product is a core. We then present a method of transforming the example into a family of graphs.